The Exponential Transmuted Exponential distribution (ETE) appeared in and in this paper we present a new generalization of the ETE distribution based on an application of the Ampadu-G family of distributions that appeared in [1,2]. We also show the new distribution is a good fit to some real-life data, indicating practical significance. As a further development, we propose a new class of distributions based on the structure of the weight function introduced in [3].
At first we recall the following definitions
Definition 1.1.
Let λ > 0, ξ > 0 be a parameter vector all of whose entries are positive, and x ? R [2]. A random variable X will be said to follow the Ampadu-G family of distributions if the CDF is given by
And the PDF is given by
Where the baseline distribution has CDF G(x; ξ) and PDF g(x; ξ)
Definition 1.2.
Assume the random variable T with support [0,∞) has CDF G(t; ξ) and PDF g(t; ξ) [2]. We say a random variable S is A
T - X(W) distributed of type II if the CDF can be expressed as either one of the following integrals
or
Where λ, ξ > 0, and the random variable X with parameter vector ω has CDF F(x; ω) and PDF f(x; ω)
As an application we consider the data on patients with breast cancer [1]. We assume the random variable T is exponentially distributed. So that the CDF of T is given by
G(t; d) = 1 - e-dt
For t, d > 0 and the PDF is given by
g(t; d) = de-dt
For t, d > 0. Now we recall the following, for some baseline distribution K(x), the transmuted family of distributions has CDF given by
(1 + b)K(x) - bK2(x)
Where b ? [-1, 1]. Now we assume X is transmuted exponentially distributed, so that if
K(x):= 1 - e-ax
For x, a > 0, then the CDF of X is given by
F(x; a, b) = (b + 1) (1 - e-ax) - b (1 - e-ax)2
And the PDF is given by
f(x; a, b) = a(b + 1) e-ax - 2abe-ax (1 - e-ax)
Now from the first integral in 3.2 Definition, we have the following
Theorem 2.1.
The CDF of the A
{Exponential}-{Transmuted Exponential} distribution is given by
Where x, a, c, d > 0 and b ? [-1, 1]
Remark 2.2.
If a random variable Q has CDF given by the above theorem, write
Q ~ AETE(a, b, c, d)
The PDF of the AETE distribution can be obtained by differentiating the CDF
The AETE distribution is seen to be a good fit to real life data as shown in the figure 1.
Figure 1: The CDF of AETE(0.0133756, 0.216229, 3.93753, 0.910041) fitted to the empirical distribution of the data on patients with breast cancer [1].
Inspired by the structure of the weight function introduced in, we ask the reader to investigate some properties and applications of a so-called A{New T} - X(W) family of distributions of type II [3]. We leave the reader with the following.
Definition 3.1.
Assume the random variable T with support [0,∞) has CDF G(t; ξ) and PDF g(t; ξ). We say a random variable S
New is A
{New T} - X(W) distributed of type II if the CDF can be expressed as the following integral.
Where λ, ξ > 0, and the random variable X with parameter vector ω has CDF F(x; ω) and PDF f(x; ω)
Our hope is that the new class of distributions presented in this paper will find application in cancer modeling and forecasting
